The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

Trigonometric Cesàro bases in the spaces of functions integrable with power weight

(2), k1, >0, L _p (0,), 1p L =C. , , p, k, (C, )- L _p (0,), , , {sinnx} _n ^=k/ (C, )- L _p (0,) |x|^p . , 1p, {x sinnx} _n=k , k2 2k–2–1/p<2k–1/p, (C, )- L ^p [0,] , >–(p–1)/.     

Нормаляный поридок аддитявных арифметичыских функций на множестве дсдвинутых” пностых чясел

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Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

О решеткахИ-Радикалов, строгих слева Радикалов, наследственных слева радикалов

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On functions having Cauchy differences in some prescribed sets

Summary AssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK (6C) = {0}. If a functionh:E F is continuous at at least one point andh(x + y) – h(x) – h(y) K + C for allx, y E, then there exists a continuous linear functiona:E F such thath(x) – a(x) K + C for everyx E.     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Les classes de Chern des représentations galoisiennes complexes

Let F be a field, G _F its absolute Galois group, : G _FGL(C) a continuous complex representation of G _F and c _i() H_2i(F, Z) its Chern classes. We show, under a mild assumption on F. that c _i ()=0 for all i2. For general F, one has that 2c_i ()=0 for all i 2.

Cette dernière condition résulte en fait de la continuité de .     

Perfect splines of least uniform deviation

(v _k) ₁ ⁿ , 1v _kr, v₁+...+v_n r, p(;r) — =(x ₁,v ₁),...,(x _n,v _n), v _1+...+v _n-r ^(r(;·)=1. , (^*;t) [, ] (;t) _{x 1}<... n v ₁ ...,v _n. (^*;t) .     

On Ω-closed sets and Ωs-closed sets in topological spaces

Summary New classes of sets called -closed sets and s-closed sets are introduced and studied. Also, we introduce and study -continuous functions and s-continuous functions and prove pasting lemma for these functions. Moreover, we introduce classes of topological spaces -T_1/2 and -T_s.     

Volumes des représentations sur un corps local

Let be a locally compact second countable group, F a local field of characteristic zero and G an F-almost-simple F-algebraic group. In this paper we study the space X(,G) of Zariski-dense representations : G = G(F) using the natural morphism of cohomological functors * : H*(G, ·) H*(, ·) (where H denotes the continuous cohomology).First let F be a p-adic field. We completely describe the relations between the geometry and the cohomology of G : using geometric properties of the Bruhat-Tits building of G we construct natural cocycles for any irreducible cohomological representation of G. We then adapt these results to the case where the field F is archimedean.Using these cocycles we obtain a simple cohomological characterization of representations with bounded image.Our main result is then the construction, using the previous cocycles and dynamical properties at infinity of , of cohomological invariants (called volumes) on the space X(,G). These volumes describe how the image () goes to infinity in G. They have coefficients in the natural universal infinite-dimensional representation L(, )$\mathbb{C}$ of .In the case where is a cocompact lattice of SO(n, 1) or SU(n, 1), we use these volumes to produce new non-trivial numerical invariants on X(,G), which refine previously known invariants.

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Volumes des représentations sur un corps local

     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

Generalized localization and convergence tests for double trigonometric Fourier series of functions fromL p,p>1

. E , f(x)L _p (T ^N ),P1,f(x)=0 E (E— ^N =[-, ]^N) E , . , .

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In closing the author thanks V. A. Il'in and . A. Alimov for their constant attention paid to the present work.     

Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

On the dimensions of automorphism groups of eight-dimensional ternary fields,II

LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF of is connected, then is either isomorphic to one of the compact Lie groupsG ₂ or SU₃, or the (covering) dimension of is at most 7.     

On the homogeneity of additive mappings

LetK be a ring with an identity 1 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M L, the setH _f :={ K f(x)=f(x) for allx M} forms a subring ofK, the homogeneity ring off. It is shown that, forM {0},L {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:ML such thatH _f =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M M makingM into a ring but not into aK-algebra.